Spatial Indexing

1
Spatial Indexing

• SAMs

2
Spatial Access Methods

• PAMs
• Grid File
• kd-tree based (LSD-, hB- trees)
• Z-ordering B-tree
• R-tree
• Variations R-tree, Hilbert R-tree

3
R-tree
Multi-way external memory structure, indexes
MBRs Dynamic structure
P1
P3
I
C
A
G
H
F
B
J
E
P4
D
P2
4
R-tree properties

• Main points
• every parent node completely covers its
  children
• a child MBR may be covered by more than one
  parent - it is stored under ONLY ONE of them.
  (ie., no need for dup. elim.)
• a point query may follow multiple branches.

5
R-tree

• The original R-tree tries to minimize the area of
  each enclosing rectangle in the index nodes.
• Is there any other property that can be
  optimized?

R-tree ? Yes!
6
R-tree

• Optimization Criteria
• (O1) Area covered by an index MBR
• (O2) Overlap between directory MBRs
• (O3) Margin of a directory rectangle
• (O4) Storage utilization
• Sometimes it is impossible to optimize all the
  above criteria at the same time!

7
R-tree

• ChooseSubtree
• If next node is a leaf node, choose the node
  using the following criteria
• Least overlap enlargement
• Least area enlargement
• Smaller area
• Else
• Least area enlargement
• Smaller area

8
R-tree

• SplitNode
• Choose the axis to split
• Choose the two groups along the chosen axis
• ChooseSplitAxis
• Along each axis, sort rectangles and break them
  into two groups (M-2m2 possible ways where one
  group contains at least m rectangles). Compute
  the sum S of all margin-values (perimeters) of
  each pair of groups. Choose the one that
  minimizes S
• ChooseSplitIndex
• Along the chosen axis, choose the grouping that
  gives the minimum overlap-value

9
R-tree

• Forced Reinsert
• defer splits, by forced-reinsert, i.e. instead
  of splitting, temporarily delete some entries,
  shrink overflowing MBR, and re-insert those
  entries
• Which ones to re-insert?
• How many? A 30

10
R-tree variations

• What about static datasets?
• (no ins/del) Hilbert
• What about other bounding shapes?

11
R-trees - variations

• what about static datasets (no ins/del/upd)?
• Q Best way to pack points?

12
R-trees - variations

• what about static datasets (no ins/del/upd)?
• Q Best way to pack points?
• A1 plane-sweep
• great for queries on x
• terrible for y

13
R-trees - variations

• what about static datasets (no ins/del/upd)?
• Q Best way to pack points?
• A1 plane-sweep
• great for queries on x
• bad for y

14
R-trees - variations

• what about static datasets (no ins/del/upd)?
• Q Best way to pack points?
• A1 plane-sweep
• great for queries on x
• terrible for y
• Q how to improve?

15
R-trees - variations

• A plane-sweep on HILBERT curve!

16
R-trees - variations

• A plane-sweep on HILBERT curve!
• In fact, it can be made dynamic (how?), as well
  as to handle regions (how?)

17
R-trees - variations

• Dynamic (Hilbert R-tree)
• each point has an h-value (hilbert value)
• insertions like a B-tree on the h-value
• but also store MBR, for searches

18
Hilbert R-tree

• Data structure of a node?

x-low, ylow x-high, y-high
LHV
ptr
h-value gt LHV MBRs inside parent MBR
19
R-trees - variations

• Data structure of a node?

B-tree
x-low, ylow x-high, y-high
LHV
ptr
h-value gt LHV MBRs inside parent MBR
20
R-trees - variations

• Data structure of a node?

R-tree
x-low, ylow x-high, y-high
LHV
ptr
h-value gt LHV MBRs inside parent MBR